Fluctuations of the Traces of Complex-Valued Random Matrices

The aim of this paper is to provide a central limit theorem for complex random matrices (X-i,(j))(i),(j >= 1) with i. i. d. entries having moments of any order. Tao and Vu (Ann. Probab. 38(5): 2023-2065, 2010) showed that for large renormalized random matrices, the spectral measure converges to a circular law. Rider and Silverstein (Ann. Probab. 34(6): 2118-2143, 2006) studied the fluctuations around this circular law in the case where the imaginary part and the real part of the random variable Xi; j have densities with respect to Lebesgue measure which have an upper bound, and their moments of order k do not grow faster than k(ak), with alpha > 0. Their result does not cover the case of real random matrices. Nourdin and Peccati (ALEA 7: 341-375, 2008) established a central limit theorem for real random matrices using a probabilistic approach. The main contribution of this paper is to use the same probabilistic approach to generalize the central limit theorem to complex random matrices.